Two functional inequalities

$\textbf{Problem 1.}$ Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x$ and $y$ holds

$$f(x)f(x+y)\ge f(x)^2+xy.$$

$\textbf{Problem 2.}$ Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ with $f(0)=0,\ f(1)=1$ and such that for all $x$ and $y$ holds

$$f(x+y)\ge 2022^xf(x)+f(y).$$